#444319

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,3)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,28,24,30,23,29)(19,31,32,27,20,26,21,25);(2,3)(9,11,12,16)(10,13,14,15)(17,19,24,20)(18,25,30,27)(21,23,32,22)(26,29,31,28);(10,14)(13,15)(17,20,23,32,24,19,22,21)(18,26,29,27,30,31,28,25);(1,3,4,2)(6,8)(11,16)(13,15)(17,21,22,19,24,32,23,20)(18,27,28,26,30,25,29,31)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=g,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351515);